Answers to Example Problems

Do various occupational groups differ in their diets? In a British study
of this question, two of the groups compared were 98 drivers and 83 conductors
of London double-decker buses. The article reporting the study gives the data
as "Mean daily consumption (± s. e.)." Some of the study
results appear below. [From Marr and Heady, Human Nutrition: Applied Nutrition
40A (1986), pp. 347-364.]

Drivers

Conductors

Total calories

2821 ± 44

2844 ± 48

Alcohol (g)

0.24 ± 0.06

0.39 ± 0.11

(a) What does "s. e." stand for? Give the mean and standard deviation
for each of the four sets of measurements.

"s. e." is the standard error. Since they have given
you the standard error, you have to do some calculation if you want the
standard deviation. You have to multiply the standard error by the square
root of the sample size to get the standard deviation. In other words, they
could have provided the following table:

Drivers

Conductors

Total calories (mean ±
sd)

2821 ± 436

2844 ± 437

Alcohol (mean ± sd)
(g)

0.24 ± 0.59

0.39 ± 1.00

(b) Is there significant evidence at the 5% level that conductors consume
more calories per day than do drivers?

The easiest way to do this is to look at the original table with
standard errors and note that the difference in between Drivers and Conductors
is only 23 calories and that this difference is much less than even one
standard error in either sample. There is no way this will result in a significant
difference from a significance test.

If you want to actually do a t-test, you just divide the
difference by the pooled standard error (see p. 169 in The Cartoon Guide).
In this case, the pooled standard error is 65, so the t-statistic
would be 23/65 = 0.35. This result needs to be greater than 2 for us to
conclude that there is a significant difference between the two populations.
(To get the pooled standard error, square each individual standard error,
add them, and then take the square root.)

(c) How significant is the difference in alcohol consumption between the
two groups? Give either a P value or some equivalent statistic.

Using the same method as described in part (b), we can find a t-statistic.
The pooled standard error is 0.125; the difference between the means is
0.15. The t-statistic is 0.15/0.125 = 1.20. Again, this does not
reach our 95% confidence level. If you use a t-table or Excel to
find the P value, it should come out near P = 0.12. This
means that we only have evidence of significant difference to the 88% confidence
level.

There are four major blood types in humans: O, A, B, and AB. In a study
conducted using blood samples from the Blood Bank of Hawaii in 1950, individuals
were classified according to blood type and ethnic group. Summarize the data.
Is there evidence to conclude that blood type and ethnic group are related?
Explain how you arrive at your conclusion.

Blood Type

Hawaiian

Hawaiian-white

Hawaiian-Chinese

White

O

1903

4469

2206

53,759

A

2490

4671

2368

50,008

B

178

606

568

16,252

AB

99

236

243

5001

The first step is to convert the above table to a frequency table
showing the frequency of each blood type within each population.

Blood Type

Hawaiian
n=4670
ME ~ 0.014

Hawaiian-white
n=9982
ME ~ 0.010

Hawaiian-Chinese
n=5385
ME ~ 0.013

White
n=125,020
ME ~ 0.003

O

0.407

0.448

0.410

0.430

A

0.533

0.468

0.440

0.400

B

0.038

0.061

0.105

0.130

AB

0.021

0.024

0.045

0.040

where n is the total sample size for that population and ME is the
margin of error. The margin of error is just twice the standard error assuming
a frequency of 0.5 ( sqrt[ 0.5 (1-0.5) / n ] ), which should be fine for looking
at the O and A blood types. For B and AB types, we should make adjustments
for the small frequencies, this would make the margins of error about half
as big.

Clearly there are areas of significant difference. Hawiians have
significantly higher rates of blood type A compared to all other populations
surveyed. Hawaiians and Hawaiian-Chinese have significantly lower rates of
blood type O compared to the other groups. What is interesting is that the
rate of blood type O is significantly higher in Hawaiian-whites compared to
either Hawaiians or Whites. Hawaiian-Chinese appear to have lower rates of
blood types O and A and higher rates of B and AB. Whites have the lowest rate
of blood type A, significantly lower than all other groups; and they have
higher rates of O and B blood types.

There are more specific tools for analyzing the differences in these
data, but general comments like these are sufficient for this program.

According to the National Center for Education Statistics, the average
base salary for the 1999-2000 school year for male full-time teachers in public
elementary and secondary school was $41,104 while the average base salary
for female full-time teachers was $39,475. The standard error on each figure
was reported as $198 and $126, respectively. Are the base salaries of male
teachers significantly different from female teachers? Given that most public
school salaries are set by a pay scale which assigns salaries by years of
education and years of experience and which makes no reference to gender,
can you come up with any reason for the gender difference in average salaries?

Using the same methods as used above in #1, you can see that the
difference between the salaries is $1,629 which is much more than two or
three or even five times either of the standard errors given. So the difference
in base salaries is clearly significant. In fact, a t-test would yield a
t-statistic of about 1629 / 235 = 7. This is way past 2 standard errors
of difference. But public schools set salaries based only on years of experience
and years of education. Why is there a significant difference based on gender?
It must be the case that, on average, males have more experience and/or
more education. There is one other factor. Usually middle and high school
teachers get paid more than elementary school teachers. So another hypothesis
would be that a greater percentage of male teachers are middle and high
school teachers as compared to the percentages for female teachers.